# Properties

 Label 2175.2.c.o.349.12 Level $2175$ Weight $2$ Character 2175.349 Analytic conductor $17.367$ Analytic rank $0$ Dimension $14$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} + 24x^{12} + 224x^{10} + 1023x^{8} + 2364x^{6} + 2612x^{4} + 1241x^{2} + 196$$ x^14 + 24*x^12 + 224*x^10 + 1023*x^8 + 2364*x^6 + 2612*x^4 + 1241*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 349.12 Root $$2.26695i$$ of defining polynomial Character $$\chi$$ $$=$$ 2175.349 Dual form 2175.2.c.o.349.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.26695i q^{2} -1.00000i q^{3} -3.13907 q^{4} +2.26695 q^{6} +0.159887i q^{7} -2.58223i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+2.26695i q^{2} -1.00000i q^{3} -3.13907 q^{4} +2.26695 q^{6} +0.159887i q^{7} -2.58223i q^{8} -1.00000 q^{9} -3.24930 q^{11} +3.13907i q^{12} -7.05679i q^{13} -0.362456 q^{14} -0.424361 q^{16} +7.63808i q^{17} -2.26695i q^{18} +3.99028 q^{19} +0.159887 q^{21} -7.36601i q^{22} -2.06402i q^{23} -2.58223 q^{24} +15.9974 q^{26} +1.00000i q^{27} -0.501897i q^{28} +1.00000 q^{29} +10.2044 q^{31} -6.12646i q^{32} +3.24930i q^{33} -17.3152 q^{34} +3.13907 q^{36} -4.27815i q^{37} +9.04576i q^{38} -7.05679 q^{39} -5.12330 q^{41} +0.362456i q^{42} +3.94180i q^{43} +10.1998 q^{44} +4.67903 q^{46} +5.61626i q^{47} +0.424361i q^{48} +6.97444 q^{49} +7.63808 q^{51} +22.1518i q^{52} -12.0822i q^{53} -2.26695 q^{54} +0.412864 q^{56} -3.99028i q^{57} +2.26695i q^{58} +7.06781 q^{59} +11.4310 q^{61} +23.1329i q^{62} -0.159887i q^{63} +13.0397 q^{64} -7.36601 q^{66} -14.9512i q^{67} -23.9765i q^{68} -2.06402 q^{69} +3.43347 q^{71} +2.58223i q^{72} +12.0772i q^{73} +9.69836 q^{74} -12.5258 q^{76} -0.519521i q^{77} -15.9974i q^{78} +12.7441 q^{79} +1.00000 q^{81} -11.6143i q^{82} +2.59792i q^{83} -0.501897 q^{84} -8.93586 q^{86} -1.00000i q^{87} +8.39044i q^{88} -4.33165 q^{89} +1.12829 q^{91} +6.47910i q^{92} -10.2044i q^{93} -12.7318 q^{94} -6.12646 q^{96} -3.88355i q^{97} +15.8107i q^{98} +3.24930 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 20 q^{4} + 4 q^{6} - 14 q^{9}+O(q^{10})$$ 14 * q - 20 * q^4 + 4 * q^6 - 14 * q^9 $$14 q - 20 q^{4} + 4 q^{6} - 14 q^{9} + 8 q^{11} - 30 q^{14} + 24 q^{16} - 30 q^{19} - 2 q^{21} - 6 q^{24} + 12 q^{26} + 14 q^{29} + 10 q^{31} - 14 q^{34} + 20 q^{36} + 2 q^{39} + 44 q^{41} - 30 q^{44} - 8 q^{46} - 24 q^{49} + 16 q^{51} - 4 q^{54} + 28 q^{56} - 12 q^{59} + 46 q^{61} - 10 q^{64} + 6 q^{66} - 28 q^{69} + 52 q^{71} + 20 q^{74} + 92 q^{76} - 28 q^{79} + 14 q^{81} + 48 q^{84} + 88 q^{86} - 28 q^{89} + 26 q^{91} + 6 q^{94} + 36 q^{96} - 8 q^{99}+O(q^{100})$$ 14 * q - 20 * q^4 + 4 * q^6 - 14 * q^9 + 8 * q^11 - 30 * q^14 + 24 * q^16 - 30 * q^19 - 2 * q^21 - 6 * q^24 + 12 * q^26 + 14 * q^29 + 10 * q^31 - 14 * q^34 + 20 * q^36 + 2 * q^39 + 44 * q^41 - 30 * q^44 - 8 * q^46 - 24 * q^49 + 16 * q^51 - 4 * q^54 + 28 * q^56 - 12 * q^59 + 46 * q^61 - 10 * q^64 + 6 * q^66 - 28 * q^69 + 52 * q^71 + 20 * q^74 + 92 * q^76 - 28 * q^79 + 14 * q^81 + 48 * q^84 + 88 * q^86 - 28 * q^89 + 26 * q^91 + 6 * q^94 + 36 * q^96 - 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.26695i 1.60298i 0.598010 + 0.801489i $$0.295958\pi$$
−0.598010 + 0.801489i $$0.704042\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −3.13907 −1.56954
$$5$$ 0 0
$$6$$ 2.26695 0.925480
$$7$$ 0.159887i 0.0604316i 0.999543 + 0.0302158i $$0.00961945\pi$$
−0.999543 + 0.0302158i $$0.990381\pi$$
$$8$$ − 2.58223i − 0.912955i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.24930 −0.979701 −0.489851 0.871806i $$-0.662949\pi$$
−0.489851 + 0.871806i $$0.662949\pi$$
$$12$$ 3.13907i 0.906173i
$$13$$ − 7.05679i − 1.95720i −0.205770 0.978600i $$-0.565970\pi$$
0.205770 0.978600i $$-0.434030\pi$$
$$14$$ −0.362456 −0.0968704
$$15$$ 0 0
$$16$$ −0.424361 −0.106090
$$17$$ 7.63808i 1.85251i 0.376901 + 0.926254i $$0.376990\pi$$
−0.376901 + 0.926254i $$0.623010\pi$$
$$18$$ − 2.26695i − 0.534326i
$$19$$ 3.99028 0.915432 0.457716 0.889098i $$-0.348668\pi$$
0.457716 + 0.889098i $$0.348668\pi$$
$$20$$ 0 0
$$21$$ 0.159887 0.0348902
$$22$$ − 7.36601i − 1.57044i
$$23$$ − 2.06402i − 0.430377i −0.976572 0.215189i $$-0.930963\pi$$
0.976572 0.215189i $$-0.0690366\pi$$
$$24$$ −2.58223 −0.527095
$$25$$ 0 0
$$26$$ 15.9974 3.13735
$$27$$ 1.00000i 0.192450i
$$28$$ − 0.501897i − 0.0948496i
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 10.2044 1.83277 0.916383 0.400303i $$-0.131095\pi$$
0.916383 + 0.400303i $$0.131095\pi$$
$$32$$ − 6.12646i − 1.08302i
$$33$$ 3.24930i 0.565631i
$$34$$ −17.3152 −2.96953
$$35$$ 0 0
$$36$$ 3.13907 0.523179
$$37$$ − 4.27815i − 0.703323i −0.936127 0.351662i $$-0.885617\pi$$
0.936127 0.351662i $$-0.114383\pi$$
$$38$$ 9.04576i 1.46742i
$$39$$ −7.05679 −1.12999
$$40$$ 0 0
$$41$$ −5.12330 −0.800125 −0.400063 0.916488i $$-0.631012\pi$$
−0.400063 + 0.916488i $$0.631012\pi$$
$$42$$ 0.362456i 0.0559282i
$$43$$ 3.94180i 0.601118i 0.953763 + 0.300559i $$0.0971733\pi$$
−0.953763 + 0.300559i $$0.902827\pi$$
$$44$$ 10.1998 1.53768
$$45$$ 0 0
$$46$$ 4.67903 0.689885
$$47$$ 5.61626i 0.819215i 0.912262 + 0.409608i $$0.134334\pi$$
−0.912262 + 0.409608i $$0.865666\pi$$
$$48$$ 0.424361i 0.0612513i
$$49$$ 6.97444 0.996348
$$50$$ 0 0
$$51$$ 7.63808 1.06955
$$52$$ 22.1518i 3.07190i
$$53$$ − 12.0822i − 1.65962i −0.558049 0.829808i $$-0.688450\pi$$
0.558049 0.829808i $$-0.311550\pi$$
$$54$$ −2.26695 −0.308493
$$55$$ 0 0
$$56$$ 0.412864 0.0551713
$$57$$ − 3.99028i − 0.528525i
$$58$$ 2.26695i 0.297665i
$$59$$ 7.06781 0.920151 0.460075 0.887880i $$-0.347822\pi$$
0.460075 + 0.887880i $$0.347822\pi$$
$$60$$ 0 0
$$61$$ 11.4310 1.46359 0.731796 0.681524i $$-0.238682\pi$$
0.731796 + 0.681524i $$0.238682\pi$$
$$62$$ 23.1329i 2.93788i
$$63$$ − 0.159887i − 0.0201439i
$$64$$ 13.0397 1.62996
$$65$$ 0 0
$$66$$ −7.36601 −0.906694
$$67$$ − 14.9512i − 1.82658i −0.407311 0.913289i $$-0.633534\pi$$
0.407311 0.913289i $$-0.366466\pi$$
$$68$$ − 23.9765i − 2.90758i
$$69$$ −2.06402 −0.248478
$$70$$ 0 0
$$71$$ 3.43347 0.407478 0.203739 0.979025i $$-0.434691\pi$$
0.203739 + 0.979025i $$0.434691\pi$$
$$72$$ 2.58223i 0.304318i
$$73$$ 12.0772i 1.41353i 0.707449 + 0.706764i $$0.249846\pi$$
−0.707449 + 0.706764i $$0.750154\pi$$
$$74$$ 9.69836 1.12741
$$75$$ 0 0
$$76$$ −12.5258 −1.43680
$$77$$ − 0.519521i − 0.0592049i
$$78$$ − 15.9974i − 1.81135i
$$79$$ 12.7441 1.43383 0.716913 0.697163i $$-0.245554\pi$$
0.716913 + 0.697163i $$0.245554\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 11.6143i − 1.28258i
$$83$$ 2.59792i 0.285159i 0.989783 + 0.142580i $$0.0455397\pi$$
−0.989783 + 0.142580i $$0.954460\pi$$
$$84$$ −0.501897 −0.0547614
$$85$$ 0 0
$$86$$ −8.93586 −0.963579
$$87$$ − 1.00000i − 0.107211i
$$88$$ 8.39044i 0.894423i
$$89$$ −4.33165 −0.459154 −0.229577 0.973290i $$-0.573734\pi$$
−0.229577 + 0.973290i $$0.573734\pi$$
$$90$$ 0 0
$$91$$ 1.12829 0.118277
$$92$$ 6.47910i 0.675493i
$$93$$ − 10.2044i − 1.05815i
$$94$$ −12.7318 −1.31318
$$95$$ 0 0
$$96$$ −6.12646 −0.625279
$$97$$ − 3.88355i − 0.394315i −0.980372 0.197158i $$-0.936829\pi$$
0.980372 0.197158i $$-0.0631711\pi$$
$$98$$ 15.8107i 1.59712i
$$99$$ 3.24930 0.326567
$$100$$ 0 0
$$101$$ 15.4045 1.53280 0.766400 0.642363i $$-0.222046\pi$$
0.766400 + 0.642363i $$0.222046\pi$$
$$102$$ 17.3152i 1.71446i
$$103$$ − 7.68484i − 0.757210i −0.925558 0.378605i $$-0.876404\pi$$
0.925558 0.378605i $$-0.123596\pi$$
$$104$$ −18.2222 −1.78684
$$105$$ 0 0
$$106$$ 27.3897 2.66033
$$107$$ − 4.60924i − 0.445592i −0.974865 0.222796i $$-0.928482\pi$$
0.974865 0.222796i $$-0.0715183\pi$$
$$108$$ − 3.13907i − 0.302058i
$$109$$ 14.4509 1.38415 0.692073 0.721827i $$-0.256698\pi$$
0.692073 + 0.721827i $$0.256698\pi$$
$$110$$ 0 0
$$111$$ −4.27815 −0.406064
$$112$$ − 0.0678498i − 0.00641120i
$$113$$ 4.08614i 0.384392i 0.981357 + 0.192196i $$0.0615610\pi$$
−0.981357 + 0.192196i $$0.938439\pi$$
$$114$$ 9.04576 0.847213
$$115$$ 0 0
$$116$$ −3.13907 −0.291456
$$117$$ 7.05679i 0.652400i
$$118$$ 16.0224i 1.47498i
$$119$$ −1.22123 −0.111950
$$120$$ 0 0
$$121$$ −0.442038 −0.0401853
$$122$$ 25.9136i 2.34610i
$$123$$ 5.12330i 0.461952i
$$124$$ −32.0324 −2.87659
$$125$$ 0 0
$$126$$ 0.362456 0.0322901
$$127$$ − 3.69650i − 0.328011i −0.986459 0.164006i $$-0.947559\pi$$
0.986459 0.164006i $$-0.0524415\pi$$
$$128$$ 17.3074i 1.52977i
$$129$$ 3.94180 0.347056
$$130$$ 0 0
$$131$$ −9.92811 −0.867423 −0.433712 0.901052i $$-0.642796\pi$$
−0.433712 + 0.901052i $$0.642796\pi$$
$$132$$ − 10.1998i − 0.887779i
$$133$$ 0.637993i 0.0553210i
$$134$$ 33.8936 2.92797
$$135$$ 0 0
$$136$$ 19.7233 1.69126
$$137$$ 3.82810i 0.327057i 0.986539 + 0.163528i $$0.0522875\pi$$
−0.986539 + 0.163528i $$0.947712\pi$$
$$138$$ − 4.67903i − 0.398305i
$$139$$ 2.06987 0.175564 0.0877822 0.996140i $$-0.472022\pi$$
0.0877822 + 0.996140i $$0.472022\pi$$
$$140$$ 0 0
$$141$$ 5.61626 0.472974
$$142$$ 7.78351i 0.653177i
$$143$$ 22.9296i 1.91747i
$$144$$ 0.424361 0.0353634
$$145$$ 0 0
$$146$$ −27.3784 −2.26585
$$147$$ − 6.97444i − 0.575242i
$$148$$ 13.4294i 1.10389i
$$149$$ 1.46609 0.120107 0.0600536 0.998195i $$-0.480873\pi$$
0.0600536 + 0.998195i $$0.480873\pi$$
$$150$$ 0 0
$$151$$ −20.1279 −1.63799 −0.818995 0.573801i $$-0.805468\pi$$
−0.818995 + 0.573801i $$0.805468\pi$$
$$152$$ − 10.3038i − 0.835748i
$$153$$ − 7.63808i − 0.617502i
$$154$$ 1.17773 0.0949041
$$155$$ 0 0
$$156$$ 22.1518 1.77356
$$157$$ 7.09106i 0.565928i 0.959131 + 0.282964i $$0.0913177\pi$$
−0.959131 + 0.282964i $$0.908682\pi$$
$$158$$ 28.8903i 2.29839i
$$159$$ −12.0822 −0.958180
$$160$$ 0 0
$$161$$ 0.330009 0.0260084
$$162$$ 2.26695i 0.178109i
$$163$$ 6.58715i 0.515946i 0.966152 + 0.257973i $$0.0830545\pi$$
−0.966152 + 0.257973i $$0.916945\pi$$
$$164$$ 16.0824 1.25583
$$165$$ 0 0
$$166$$ −5.88937 −0.457104
$$167$$ − 10.6638i − 0.825192i −0.910914 0.412596i $$-0.864622\pi$$
0.910914 0.412596i $$-0.135378\pi$$
$$168$$ − 0.412864i − 0.0318532i
$$169$$ −36.7983 −2.83063
$$170$$ 0 0
$$171$$ −3.99028 −0.305144
$$172$$ − 12.3736i − 0.943477i
$$173$$ 13.8799i 1.05527i 0.849472 + 0.527633i $$0.176920\pi$$
−0.849472 + 0.527633i $$0.823080\pi$$
$$174$$ 2.26695 0.171857
$$175$$ 0 0
$$176$$ 1.37888 0.103937
$$177$$ − 7.06781i − 0.531249i
$$178$$ − 9.81965i − 0.736014i
$$179$$ 11.1105 0.830439 0.415219 0.909721i $$-0.363705\pi$$
0.415219 + 0.909721i $$0.363705\pi$$
$$180$$ 0 0
$$181$$ 10.9453 0.813557 0.406778 0.913527i $$-0.366652\pi$$
0.406778 + 0.913527i $$0.366652\pi$$
$$182$$ 2.55777i 0.189595i
$$183$$ − 11.4310i − 0.845005i
$$184$$ −5.32976 −0.392915
$$185$$ 0 0
$$186$$ 23.1329 1.69619
$$187$$ − 24.8184i − 1.81490i
$$188$$ − 17.6298i − 1.28579i
$$189$$ −0.159887 −0.0116301
$$190$$ 0 0
$$191$$ 18.1472 1.31309 0.656544 0.754287i $$-0.272017\pi$$
0.656544 + 0.754287i $$0.272017\pi$$
$$192$$ − 13.0397i − 0.941058i
$$193$$ 17.3305i 1.24747i 0.781634 + 0.623737i $$0.214386\pi$$
−0.781634 + 0.623737i $$0.785614\pi$$
$$194$$ 8.80383 0.632078
$$195$$ 0 0
$$196$$ −21.8933 −1.56381
$$197$$ − 10.4273i − 0.742915i −0.928450 0.371458i $$-0.878858\pi$$
0.928450 0.371458i $$-0.121142\pi$$
$$198$$ 7.36601i 0.523480i
$$199$$ −11.5570 −0.819256 −0.409628 0.912253i $$-0.634341\pi$$
−0.409628 + 0.912253i $$0.634341\pi$$
$$200$$ 0 0
$$201$$ −14.9512 −1.05458
$$202$$ 34.9212i 2.45704i
$$203$$ 0.159887i 0.0112219i
$$204$$ −23.9765 −1.67869
$$205$$ 0 0
$$206$$ 17.4212 1.21379
$$207$$ 2.06402i 0.143459i
$$208$$ 2.99463i 0.207640i
$$209$$ −12.9656 −0.896850
$$210$$ 0 0
$$211$$ −21.1616 −1.45682 −0.728412 0.685140i $$-0.759741\pi$$
−0.728412 + 0.685140i $$0.759741\pi$$
$$212$$ 37.9269i 2.60483i
$$213$$ − 3.43347i − 0.235257i
$$214$$ 10.4489 0.714274
$$215$$ 0 0
$$216$$ 2.58223 0.175698
$$217$$ 1.63155i 0.110757i
$$218$$ 32.7595i 2.21876i
$$219$$ 12.0772 0.816101
$$220$$ 0 0
$$221$$ 53.9003 3.62573
$$222$$ − 9.69836i − 0.650911i
$$223$$ − 9.46925i − 0.634108i −0.948408 0.317054i $$-0.897306\pi$$
0.948408 0.317054i $$-0.102694\pi$$
$$224$$ 0.979541 0.0654483
$$225$$ 0 0
$$226$$ −9.26310 −0.616172
$$227$$ 22.4030i 1.48694i 0.668768 + 0.743471i $$0.266822\pi$$
−0.668768 + 0.743471i $$0.733178\pi$$
$$228$$ 12.5258i 0.829539i
$$229$$ −2.03117 −0.134223 −0.0671117 0.997745i $$-0.521378\pi$$
−0.0671117 + 0.997745i $$0.521378\pi$$
$$230$$ 0 0
$$231$$ −0.519521 −0.0341820
$$232$$ − 2.58223i − 0.169532i
$$233$$ − 11.0704i − 0.725249i −0.931935 0.362624i $$-0.881881\pi$$
0.931935 0.362624i $$-0.118119\pi$$
$$234$$ −15.9974 −1.04578
$$235$$ 0 0
$$236$$ −22.1864 −1.44421
$$237$$ − 12.7441i − 0.827820i
$$238$$ − 2.76847i − 0.179453i
$$239$$ 2.24098 0.144957 0.0724786 0.997370i $$-0.476909\pi$$
0.0724786 + 0.997370i $$0.476909\pi$$
$$240$$ 0 0
$$241$$ 14.4838 0.932986 0.466493 0.884525i $$-0.345517\pi$$
0.466493 + 0.884525i $$0.345517\pi$$
$$242$$ − 1.00208i − 0.0644161i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −35.8828 −2.29716
$$245$$ 0 0
$$246$$ −11.6143 −0.740499
$$247$$ − 28.1585i − 1.79168i
$$248$$ − 26.3501i − 1.67323i
$$249$$ 2.59792 0.164637
$$250$$ 0 0
$$251$$ −16.2044 −1.02281 −0.511407 0.859339i $$-0.670875\pi$$
−0.511407 + 0.859339i $$0.670875\pi$$
$$252$$ 0.501897i 0.0316165i
$$253$$ 6.70661i 0.421641i
$$254$$ 8.37978 0.525794
$$255$$ 0 0
$$256$$ −13.1557 −0.822232
$$257$$ − 3.36942i − 0.210178i −0.994463 0.105089i $$-0.966487\pi$$
0.994463 0.105089i $$-0.0335128\pi$$
$$258$$ 8.93586i 0.556323i
$$259$$ 0.684020 0.0425029
$$260$$ 0 0
$$261$$ −1.00000 −0.0618984
$$262$$ − 22.5066i − 1.39046i
$$263$$ − 1.79671i − 0.110790i −0.998465 0.0553950i $$-0.982358\pi$$
0.998465 0.0553950i $$-0.0176418\pi$$
$$264$$ 8.39044 0.516396
$$265$$ 0 0
$$266$$ −1.44630 −0.0886783
$$267$$ 4.33165i 0.265093i
$$268$$ 46.9329i 2.86688i
$$269$$ 0.117055 0.00713699 0.00356849 0.999994i $$-0.498864\pi$$
0.00356849 + 0.999994i $$0.498864\pi$$
$$270$$ 0 0
$$271$$ −4.06294 −0.246806 −0.123403 0.992357i $$-0.539381\pi$$
−0.123403 + 0.992357i $$0.539381\pi$$
$$272$$ − 3.24131i − 0.196533i
$$273$$ − 1.12829i − 0.0682871i
$$274$$ −8.67813 −0.524265
$$275$$ 0 0
$$276$$ 6.47910 0.389996
$$277$$ − 12.5436i − 0.753674i −0.926279 0.376837i $$-0.877012\pi$$
0.926279 0.376837i $$-0.122988\pi$$
$$278$$ 4.69231i 0.281426i
$$279$$ −10.2044 −0.610922
$$280$$ 0 0
$$281$$ −2.83163 −0.168921 −0.0844605 0.996427i $$-0.526917\pi$$
−0.0844605 + 0.996427i $$0.526917\pi$$
$$282$$ 12.7318i 0.758167i
$$283$$ − 28.2249i − 1.67780i −0.544288 0.838898i $$-0.683200\pi$$
0.544288 0.838898i $$-0.316800\pi$$
$$284$$ −10.7779 −0.639551
$$285$$ 0 0
$$286$$ −51.9804 −3.07366
$$287$$ − 0.819148i − 0.0483528i
$$288$$ 6.12646i 0.361005i
$$289$$ −41.3403 −2.43178
$$290$$ 0 0
$$291$$ −3.88355 −0.227658
$$292$$ − 37.9112i − 2.21859i
$$293$$ − 21.9337i − 1.28138i −0.767799 0.640691i $$-0.778648\pi$$
0.767799 0.640691i $$-0.221352\pi$$
$$294$$ 15.8107 0.922100
$$295$$ 0 0
$$296$$ −11.0472 −0.642103
$$297$$ − 3.24930i − 0.188544i
$$298$$ 3.32357i 0.192529i
$$299$$ −14.5653 −0.842335
$$300$$ 0 0
$$301$$ −0.630241 −0.0363265
$$302$$ − 45.6291i − 2.62566i
$$303$$ − 15.4045i − 0.884963i
$$304$$ −1.69332 −0.0971185
$$305$$ 0 0
$$306$$ 17.3152 0.989843
$$307$$ − 0.461574i − 0.0263434i −0.999913 0.0131717i $$-0.995807\pi$$
0.999913 0.0131717i $$-0.00419281\pi$$
$$308$$ 1.63081i 0.0929243i
$$309$$ −7.68484 −0.437175
$$310$$ 0 0
$$311$$ 27.5507 1.56226 0.781128 0.624371i $$-0.214645\pi$$
0.781128 + 0.624371i $$0.214645\pi$$
$$312$$ 18.2222i 1.03163i
$$313$$ − 14.9121i − 0.842882i −0.906856 0.421441i $$-0.861524\pi$$
0.906856 0.421441i $$-0.138476\pi$$
$$314$$ −16.0751 −0.907170
$$315$$ 0 0
$$316$$ −40.0048 −2.25044
$$317$$ − 22.9266i − 1.28769i −0.765157 0.643843i $$-0.777339\pi$$
0.765157 0.643843i $$-0.222661\pi$$
$$318$$ − 27.3897i − 1.53594i
$$319$$ −3.24930 −0.181926
$$320$$ 0 0
$$321$$ −4.60924 −0.257263
$$322$$ 0.748115i 0.0416908i
$$323$$ 30.4781i 1.69584i
$$324$$ −3.13907 −0.174393
$$325$$ 0 0
$$326$$ −14.9328 −0.827049
$$327$$ − 14.4509i − 0.799137i
$$328$$ 13.2295i 0.730478i
$$329$$ −0.897966 −0.0495065
$$330$$ 0 0
$$331$$ −3.23147 −0.177618 −0.0888088 0.996049i $$-0.528306\pi$$
−0.0888088 + 0.996049i $$0.528306\pi$$
$$332$$ − 8.15507i − 0.447568i
$$333$$ 4.27815i 0.234441i
$$334$$ 24.1744 1.32277
$$335$$ 0 0
$$336$$ −0.0678498 −0.00370151
$$337$$ 28.8026i 1.56898i 0.620143 + 0.784489i $$0.287075\pi$$
−0.620143 + 0.784489i $$0.712925\pi$$
$$338$$ − 83.4199i − 4.53744i
$$339$$ 4.08614 0.221929
$$340$$ 0 0
$$341$$ −33.1572 −1.79556
$$342$$ − 9.04576i − 0.489139i
$$343$$ 2.23433i 0.120642i
$$344$$ 10.1786 0.548794
$$345$$ 0 0
$$346$$ −31.4650 −1.69157
$$347$$ − 16.4175i − 0.881337i −0.897670 0.440669i $$-0.854741\pi$$
0.897670 0.440669i $$-0.145259\pi$$
$$348$$ 3.13907i 0.168272i
$$349$$ 18.2107 0.974796 0.487398 0.873180i $$-0.337946\pi$$
0.487398 + 0.873180i $$0.337946\pi$$
$$350$$ 0 0
$$351$$ 7.05679 0.376663
$$352$$ 19.9067i 1.06103i
$$353$$ 18.8790i 1.00483i 0.864627 + 0.502414i $$0.167555\pi$$
−0.864627 + 0.502414i $$0.832445\pi$$
$$354$$ 16.0224 0.851581
$$355$$ 0 0
$$356$$ 13.5974 0.720660
$$357$$ 1.22123i 0.0646343i
$$358$$ 25.1870i 1.33117i
$$359$$ −14.4500 −0.762640 −0.381320 0.924443i $$-0.624530\pi$$
−0.381320 + 0.924443i $$0.624530\pi$$
$$360$$ 0 0
$$361$$ −3.07770 −0.161984
$$362$$ 24.8125i 1.30411i
$$363$$ 0.442038i 0.0232010i
$$364$$ −3.54178 −0.185640
$$365$$ 0 0
$$366$$ 25.9136 1.35452
$$367$$ − 21.1171i − 1.10231i −0.834404 0.551153i $$-0.814188\pi$$
0.834404 0.551153i $$-0.185812\pi$$
$$368$$ 0.875889i 0.0456589i
$$369$$ 5.12330 0.266708
$$370$$ 0 0
$$371$$ 1.93178 0.100293
$$372$$ 32.0324i 1.66080i
$$373$$ 17.1185i 0.886361i 0.896432 + 0.443181i $$0.146150\pi$$
−0.896432 + 0.443181i $$0.853850\pi$$
$$374$$ 56.2622 2.90925
$$375$$ 0 0
$$376$$ 14.5025 0.747907
$$377$$ − 7.05679i − 0.363443i
$$378$$ − 0.362456i − 0.0186427i
$$379$$ 0.206688 0.0106169 0.00530844 0.999986i $$-0.498310\pi$$
0.00530844 + 0.999986i $$0.498310\pi$$
$$380$$ 0 0
$$381$$ −3.69650 −0.189377
$$382$$ 41.1389i 2.10485i
$$383$$ − 16.3701i − 0.836474i −0.908338 0.418237i $$-0.862648\pi$$
0.908338 0.418237i $$-0.137352\pi$$
$$384$$ 17.3074 0.883215
$$385$$ 0 0
$$386$$ −39.2873 −1.99967
$$387$$ − 3.94180i − 0.200373i
$$388$$ 12.1908i 0.618892i
$$389$$ −11.7627 −0.596393 −0.298197 0.954504i $$-0.596385\pi$$
−0.298197 + 0.954504i $$0.596385\pi$$
$$390$$ 0 0
$$391$$ 15.7651 0.797277
$$392$$ − 18.0096i − 0.909621i
$$393$$ 9.92811i 0.500807i
$$394$$ 23.6382 1.19088
$$395$$ 0 0
$$396$$ −10.1998 −0.512559
$$397$$ − 4.96134i − 0.249002i −0.992219 0.124501i $$-0.960267\pi$$
0.992219 0.124501i $$-0.0397331\pi$$
$$398$$ − 26.1992i − 1.31325i
$$399$$ 0.637993 0.0319396
$$400$$ 0 0
$$401$$ 36.1967 1.80758 0.903789 0.427979i $$-0.140774\pi$$
0.903789 + 0.427979i $$0.140774\pi$$
$$402$$ − 33.8936i − 1.69046i
$$403$$ − 72.0103i − 3.58709i
$$404$$ −48.3557 −2.40579
$$405$$ 0 0
$$406$$ −0.362456 −0.0179884
$$407$$ 13.9010i 0.689047i
$$408$$ − 19.7233i − 0.976447i
$$409$$ 16.1607 0.799094 0.399547 0.916713i $$-0.369168\pi$$
0.399547 + 0.916713i $$0.369168\pi$$
$$410$$ 0 0
$$411$$ 3.82810 0.188826
$$412$$ 24.1233i 1.18847i
$$413$$ 1.13005i 0.0556061i
$$414$$ −4.67903 −0.229962
$$415$$ 0 0
$$416$$ −43.2331 −2.11968
$$417$$ − 2.06987i − 0.101362i
$$418$$ − 29.3924i − 1.43763i
$$419$$ 33.5544 1.63924 0.819621 0.572906i $$-0.194184\pi$$
0.819621 + 0.572906i $$0.194184\pi$$
$$420$$ 0 0
$$421$$ 26.9847 1.31515 0.657577 0.753388i $$-0.271582\pi$$
0.657577 + 0.753388i $$0.271582\pi$$
$$422$$ − 47.9723i − 2.33526i
$$423$$ − 5.61626i − 0.273072i
$$424$$ −31.1990 −1.51515
$$425$$ 0 0
$$426$$ 7.78351 0.377112
$$427$$ 1.82767i 0.0884471i
$$428$$ 14.4687i 0.699373i
$$429$$ 22.9296 1.10705
$$430$$ 0 0
$$431$$ 0.832366 0.0400937 0.0200468 0.999799i $$-0.493618\pi$$
0.0200468 + 0.999799i $$0.493618\pi$$
$$432$$ − 0.424361i − 0.0204171i
$$433$$ 22.5647i 1.08439i 0.840253 + 0.542195i $$0.182407\pi$$
−0.840253 + 0.542195i $$0.817593\pi$$
$$434$$ −3.69865 −0.177541
$$435$$ 0 0
$$436$$ −45.3625 −2.17247
$$437$$ − 8.23600i − 0.393981i
$$438$$ 27.3784i 1.30819i
$$439$$ −23.9583 −1.14347 −0.571734 0.820439i $$-0.693729\pi$$
−0.571734 + 0.820439i $$0.693729\pi$$
$$440$$ 0 0
$$441$$ −6.97444 −0.332116
$$442$$ 122.190i 5.81196i
$$443$$ − 17.4925i − 0.831092i −0.909572 0.415546i $$-0.863591\pi$$
0.909572 0.415546i $$-0.136409\pi$$
$$444$$ 13.4294 0.637332
$$445$$ 0 0
$$446$$ 21.4663 1.01646
$$447$$ − 1.46609i − 0.0693439i
$$448$$ 2.08487i 0.0985010i
$$449$$ −4.22200 −0.199249 −0.0996243 0.995025i $$-0.531764\pi$$
−0.0996243 + 0.995025i $$0.531764\pi$$
$$450$$ 0 0
$$451$$ 16.6471 0.783884
$$452$$ − 12.8267i − 0.603318i
$$453$$ 20.1279i 0.945694i
$$454$$ −50.7866 −2.38353
$$455$$ 0 0
$$456$$ −10.3038 −0.482520
$$457$$ − 1.28562i − 0.0601389i −0.999548 0.0300694i $$-0.990427\pi$$
0.999548 0.0300694i $$-0.00957284\pi$$
$$458$$ − 4.60456i − 0.215157i
$$459$$ −7.63808 −0.356515
$$460$$ 0 0
$$461$$ −23.8534 −1.11097 −0.555483 0.831528i $$-0.687466\pi$$
−0.555483 + 0.831528i $$0.687466\pi$$
$$462$$ − 1.17773i − 0.0547929i
$$463$$ − 14.1402i − 0.657152i −0.944478 0.328576i $$-0.893431\pi$$
0.944478 0.328576i $$-0.106569\pi$$
$$464$$ −0.424361 −0.0197005
$$465$$ 0 0
$$466$$ 25.0962 1.16256
$$467$$ 7.06487i 0.326923i 0.986550 + 0.163462i $$0.0522660\pi$$
−0.986550 + 0.163462i $$0.947734\pi$$
$$468$$ − 22.1518i − 1.02397i
$$469$$ 2.39050 0.110383
$$470$$ 0 0
$$471$$ 7.09106 0.326739
$$472$$ − 18.2507i − 0.840056i
$$473$$ − 12.8081i − 0.588916i
$$474$$ 28.8903 1.32698
$$475$$ 0 0
$$476$$ 3.83353 0.175710
$$477$$ 12.0822i 0.553205i
$$478$$ 5.08020i 0.232363i
$$479$$ −22.9561 −1.04889 −0.524444 0.851445i $$-0.675727\pi$$
−0.524444 + 0.851445i $$0.675727\pi$$
$$480$$ 0 0
$$481$$ −30.1900 −1.37654
$$482$$ 32.8342i 1.49556i
$$483$$ − 0.330009i − 0.0150159i
$$484$$ 1.38759 0.0630723
$$485$$ 0 0
$$486$$ 2.26695 0.102831
$$487$$ − 27.0319i − 1.22493i −0.790496 0.612467i $$-0.790177\pi$$
0.790496 0.612467i $$-0.209823\pi$$
$$488$$ − 29.5175i − 1.33619i
$$489$$ 6.58715 0.297881
$$490$$ 0 0
$$491$$ −11.5498 −0.521235 −0.260617 0.965442i $$-0.583926\pi$$
−0.260617 + 0.965442i $$0.583926\pi$$
$$492$$ − 16.0824i − 0.725051i
$$493$$ 7.63808i 0.344002i
$$494$$ 63.8340 2.87203
$$495$$ 0 0
$$496$$ −4.33036 −0.194439
$$497$$ 0.548966i 0.0246245i
$$498$$ 5.88937i 0.263909i
$$499$$ −3.77942 −0.169190 −0.0845950 0.996415i $$-0.526960\pi$$
−0.0845950 + 0.996415i $$0.526960\pi$$
$$500$$ 0 0
$$501$$ −10.6638 −0.476425
$$502$$ − 36.7346i − 1.63955i
$$503$$ 14.7684i 0.658489i 0.944245 + 0.329244i $$0.106794\pi$$
−0.944245 + 0.329244i $$0.893206\pi$$
$$504$$ −0.412864 −0.0183904
$$505$$ 0 0
$$506$$ −15.2036 −0.675881
$$507$$ 36.7983i 1.63427i
$$508$$ 11.6036i 0.514826i
$$509$$ 42.4182 1.88015 0.940077 0.340962i $$-0.110753\pi$$
0.940077 + 0.340962i $$0.110753\pi$$
$$510$$ 0 0
$$511$$ −1.93098 −0.0854217
$$512$$ 4.79143i 0.211753i
$$513$$ 3.99028i 0.176175i
$$514$$ 7.63831 0.336911
$$515$$ 0 0
$$516$$ −12.3736 −0.544717
$$517$$ − 18.2489i − 0.802586i
$$518$$ 1.55064i 0.0681312i
$$519$$ 13.8799 0.609258
$$520$$ 0 0
$$521$$ 20.3409 0.891152 0.445576 0.895244i $$-0.352999\pi$$
0.445576 + 0.895244i $$0.352999\pi$$
$$522$$ − 2.26695i − 0.0992218i
$$523$$ − 1.52190i − 0.0665481i −0.999446 0.0332740i $$-0.989407\pi$$
0.999446 0.0332740i $$-0.0105934\pi$$
$$524$$ 31.1651 1.36145
$$525$$ 0 0
$$526$$ 4.07306 0.177594
$$527$$ 77.9421i 3.39521i
$$528$$ − 1.37888i − 0.0600080i
$$529$$ 18.7398 0.814775
$$530$$ 0 0
$$531$$ −7.06781 −0.306717
$$532$$ − 2.00271i − 0.0868283i
$$533$$ 36.1540i 1.56601i
$$534$$ −9.81965 −0.424938
$$535$$ 0 0
$$536$$ −38.6074 −1.66758
$$537$$ − 11.1105i − 0.479454i
$$538$$ 0.265359i 0.0114404i
$$539$$ −22.6620 −0.976123
$$540$$ 0 0
$$541$$ −9.40679 −0.404430 −0.202215 0.979341i $$-0.564814\pi$$
−0.202215 + 0.979341i $$0.564814\pi$$
$$542$$ − 9.21049i − 0.395624i
$$543$$ − 10.9453i − 0.469707i
$$544$$ 46.7944 2.00629
$$545$$ 0 0
$$546$$ 2.55777 0.109463
$$547$$ 19.7232i 0.843304i 0.906758 + 0.421652i $$0.138550\pi$$
−0.906758 + 0.421652i $$0.861450\pi$$
$$548$$ − 12.0167i − 0.513328i
$$549$$ −11.4310 −0.487864
$$550$$ 0 0
$$551$$ 3.99028 0.169991
$$552$$ 5.32976i 0.226850i
$$553$$ 2.03762i 0.0866483i
$$554$$ 28.4358 1.20812
$$555$$ 0 0
$$556$$ −6.49749 −0.275555
$$557$$ 9.01986i 0.382184i 0.981572 + 0.191092i $$0.0612029\pi$$
−0.981572 + 0.191092i $$0.938797\pi$$
$$558$$ − 23.1329i − 0.979294i
$$559$$ 27.8164 1.17651
$$560$$ 0 0
$$561$$ −24.8184 −1.04784
$$562$$ − 6.41918i − 0.270777i
$$563$$ − 17.5283i − 0.738728i −0.929285 0.369364i $$-0.879576\pi$$
0.929285 0.369364i $$-0.120424\pi$$
$$564$$ −17.6298 −0.742350
$$565$$ 0 0
$$566$$ 63.9845 2.68947
$$567$$ 0.159887i 0.00671462i
$$568$$ − 8.86599i − 0.372009i
$$569$$ −32.3981 −1.35820 −0.679099 0.734046i $$-0.737629\pi$$
−0.679099 + 0.734046i $$0.737629\pi$$
$$570$$ 0 0
$$571$$ −11.8393 −0.495458 −0.247729 0.968829i $$-0.579684\pi$$
−0.247729 + 0.968829i $$0.579684\pi$$
$$572$$ − 71.9778i − 3.00954i
$$573$$ − 18.1472i − 0.758112i
$$574$$ 1.85697 0.0775085
$$575$$ 0 0
$$576$$ −13.0397 −0.543320
$$577$$ − 0.192778i − 0.00802545i −0.999992 0.00401273i $$-0.998723\pi$$
0.999992 0.00401273i $$-0.00127729\pi$$
$$578$$ − 93.7165i − 3.89809i
$$579$$ 17.3305 0.720229
$$580$$ 0 0
$$581$$ −0.415374 −0.0172326
$$582$$ − 8.80383i − 0.364931i
$$583$$ 39.2587i 1.62593i
$$584$$ 31.1861 1.29049
$$585$$ 0 0
$$586$$ 49.7227 2.05403
$$587$$ 34.5089i 1.42433i 0.702010 + 0.712167i $$0.252286\pi$$
−0.702010 + 0.712167i $$0.747714\pi$$
$$588$$ 21.8933i 0.902863i
$$589$$ 40.7184 1.67777
$$590$$ 0 0
$$591$$ −10.4273 −0.428922
$$592$$ 1.81548i 0.0746158i
$$593$$ − 2.42748i − 0.0996846i −0.998757 0.0498423i $$-0.984128\pi$$
0.998757 0.0498423i $$-0.0158719\pi$$
$$594$$ 7.36601 0.302231
$$595$$ 0 0
$$596$$ −4.60218 −0.188513
$$597$$ 11.5570i 0.472997i
$$598$$ − 33.0189i − 1.35024i
$$599$$ −29.3312 −1.19844 −0.599220 0.800585i $$-0.704522\pi$$
−0.599220 + 0.800585i $$0.704522\pi$$
$$600$$ 0 0
$$601$$ 29.5631 1.20590 0.602952 0.797778i $$-0.293991\pi$$
0.602952 + 0.797778i $$0.293991\pi$$
$$602$$ − 1.42873i − 0.0582306i
$$603$$ 14.9512i 0.608860i
$$604$$ 63.1831 2.57089
$$605$$ 0 0
$$606$$ 34.9212 1.41858
$$607$$ 33.1756i 1.34656i 0.739390 + 0.673278i $$0.235114\pi$$
−0.739390 + 0.673278i $$0.764886\pi$$
$$608$$ − 24.4463i − 0.991427i
$$609$$ 0.159887 0.00647894
$$610$$ 0 0
$$611$$ 39.6327 1.60337
$$612$$ 23.9765i 0.969193i
$$613$$ 13.9979i 0.565371i 0.959213 + 0.282686i $$0.0912253\pi$$
−0.959213 + 0.282686i $$0.908775\pi$$
$$614$$ 1.04637 0.0422279
$$615$$ 0 0
$$616$$ −1.34152 −0.0540514
$$617$$ − 29.9344i − 1.20511i −0.798077 0.602556i $$-0.794149\pi$$
0.798077 0.602556i $$-0.205851\pi$$
$$618$$ − 17.4212i − 0.700782i
$$619$$ 14.9032 0.599009 0.299505 0.954095i $$-0.403179\pi$$
0.299505 + 0.954095i $$0.403179\pi$$
$$620$$ 0 0
$$621$$ 2.06402 0.0828261
$$622$$ 62.4561i 2.50426i
$$623$$ − 0.692574i − 0.0277474i
$$624$$ 2.99463 0.119881
$$625$$ 0 0
$$626$$ 33.8050 1.35112
$$627$$ 12.9656i 0.517796i
$$628$$ − 22.2594i − 0.888245i
$$629$$ 32.6769 1.30291
$$630$$ 0 0
$$631$$ −28.8643 −1.14907 −0.574534 0.818480i $$-0.694817\pi$$
−0.574534 + 0.818480i $$0.694817\pi$$
$$632$$ − 32.9082i − 1.30902i
$$633$$ 21.1616i 0.841097i
$$634$$ 51.9735 2.06413
$$635$$ 0 0
$$636$$ 37.9269 1.50390
$$637$$ − 49.2171i − 1.95005i
$$638$$ − 7.36601i − 0.291623i
$$639$$ −3.43347 −0.135826
$$640$$ 0 0
$$641$$ 44.0167 1.73856 0.869278 0.494324i $$-0.164585\pi$$
0.869278 + 0.494324i $$0.164585\pi$$
$$642$$ − 10.4489i − 0.412386i
$$643$$ 29.6239i 1.16825i 0.811663 + 0.584125i $$0.198562\pi$$
−0.811663 + 0.584125i $$0.801438\pi$$
$$644$$ −1.03592 −0.0408211
$$645$$ 0 0
$$646$$ −69.0923 −2.71840
$$647$$ − 39.3943i − 1.54875i −0.632727 0.774375i $$-0.718064\pi$$
0.632727 0.774375i $$-0.281936\pi$$
$$648$$ − 2.58223i − 0.101439i
$$649$$ −22.9654 −0.901473
$$650$$ 0 0
$$651$$ 1.63155 0.0639455
$$652$$ − 20.6776i − 0.809796i
$$653$$ − 22.3518i − 0.874692i −0.899293 0.437346i $$-0.855918\pi$$
0.899293 0.437346i $$-0.144082\pi$$
$$654$$ 32.7595 1.28100
$$655$$ 0 0
$$656$$ 2.17413 0.0848855
$$657$$ − 12.0772i − 0.471176i
$$658$$ − 2.03565i − 0.0793577i
$$659$$ −42.1979 −1.64380 −0.821898 0.569634i $$-0.807085\pi$$
−0.821898 + 0.569634i $$0.807085\pi$$
$$660$$ 0 0
$$661$$ 8.38035 0.325958 0.162979 0.986630i $$-0.447890\pi$$
0.162979 + 0.986630i $$0.447890\pi$$
$$662$$ − 7.32559i − 0.284717i
$$663$$ − 53.9003i − 2.09332i
$$664$$ 6.70843 0.260337
$$665$$ 0 0
$$666$$ −9.69836 −0.375804
$$667$$ − 2.06402i − 0.0799191i
$$668$$ 33.4746i 1.29517i
$$669$$ −9.46925 −0.366102
$$670$$ 0 0
$$671$$ −37.1428 −1.43388
$$672$$ − 0.979541i − 0.0377866i
$$673$$ 3.13725i 0.120932i 0.998170 + 0.0604660i $$0.0192587\pi$$
−0.998170 + 0.0604660i $$0.980741\pi$$
$$674$$ −65.2941 −2.51504
$$675$$ 0 0
$$676$$ 115.512 4.44279
$$677$$ 16.2776i 0.625600i 0.949819 + 0.312800i $$0.101267\pi$$
−0.949819 + 0.312800i $$0.898733\pi$$
$$678$$ 9.26310i 0.355747i
$$679$$ 0.620929 0.0238291
$$680$$ 0 0
$$681$$ 22.4030 0.858486
$$682$$ − 75.1658i − 2.87825i
$$683$$ 1.78249i 0.0682050i 0.999418 + 0.0341025i $$0.0108573\pi$$
−0.999418 + 0.0341025i $$0.989143\pi$$
$$684$$ 12.5258 0.478935
$$685$$ 0 0
$$686$$ −5.06512 −0.193387
$$687$$ 2.03117i 0.0774939i
$$688$$ − 1.67275i − 0.0637728i
$$689$$ −85.2614 −3.24820
$$690$$ 0 0
$$691$$ −20.8504 −0.793187 −0.396594 0.917994i $$-0.629808\pi$$
−0.396594 + 0.917994i $$0.629808\pi$$
$$692$$ − 43.5699i − 1.65628i
$$693$$ 0.519521i 0.0197350i
$$694$$ 37.2177 1.41276
$$695$$ 0 0
$$696$$ −2.58223 −0.0978791
$$697$$ − 39.1322i − 1.48224i
$$698$$ 41.2828i 1.56258i
$$699$$ −11.0704 −0.418722
$$700$$ 0 0
$$701$$ 32.5504 1.22941 0.614707 0.788756i $$-0.289274\pi$$
0.614707 + 0.788756i $$0.289274\pi$$
$$702$$ 15.9974i 0.603783i
$$703$$ − 17.0710i − 0.643845i
$$704$$ −42.3698 −1.59687
$$705$$ 0 0
$$706$$ −42.7978 −1.61072
$$707$$ 2.46297i 0.0926295i
$$708$$ 22.1864i 0.833815i
$$709$$ −5.93195 −0.222779 −0.111390 0.993777i $$-0.535530\pi$$
−0.111390 + 0.993777i $$0.535530\pi$$
$$710$$ 0 0
$$711$$ −12.7441 −0.477942
$$712$$ 11.1853i 0.419187i
$$713$$ − 21.0621i − 0.788781i
$$714$$ −2.76847 −0.103607
$$715$$ 0 0
$$716$$ −34.8767 −1.30340
$$717$$ − 2.24098i − 0.0836911i
$$718$$ − 32.7574i − 1.22250i
$$719$$ 13.5279 0.504505 0.252252 0.967661i $$-0.418829\pi$$
0.252252 + 0.967661i $$0.418829\pi$$
$$720$$ 0 0
$$721$$ 1.22871 0.0457594
$$722$$ − 6.97701i − 0.259657i
$$723$$ − 14.4838i − 0.538660i
$$724$$ −34.3581 −1.27691
$$725$$ 0 0
$$726$$ −1.00208 −0.0371907
$$727$$ − 12.7407i − 0.472525i −0.971689 0.236263i $$-0.924077\pi$$
0.971689 0.236263i $$-0.0759226\pi$$
$$728$$ − 2.91350i − 0.107981i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −30.1078 −1.11358
$$732$$ 35.8828i 1.32627i
$$733$$ − 30.9422i − 1.14288i −0.820645 0.571438i $$-0.806386\pi$$
0.820645 0.571438i $$-0.193614\pi$$
$$734$$ 47.8716 1.76697
$$735$$ 0 0
$$736$$ −12.6451 −0.466105
$$737$$ 48.5809i 1.78950i
$$738$$ 11.6143i 0.427528i
$$739$$ −35.0047 −1.28767 −0.643834 0.765166i $$-0.722657\pi$$
−0.643834 + 0.765166i $$0.722657\pi$$
$$740$$ 0 0
$$741$$ −28.1585 −1.03443
$$742$$ 4.37926i 0.160768i
$$743$$ 12.5515i 0.460468i 0.973135 + 0.230234i $$0.0739492\pi$$
−0.973135 + 0.230234i $$0.926051\pi$$
$$744$$ −26.3501 −0.966041
$$745$$ 0 0
$$746$$ −38.8068 −1.42082
$$747$$ − 2.59792i − 0.0950530i
$$748$$ 77.9069i 2.84856i
$$749$$ 0.736956 0.0269278
$$750$$ 0 0
$$751$$ −33.8278 −1.23439 −0.617197 0.786809i $$-0.711732\pi$$
−0.617197 + 0.786809i $$0.711732\pi$$
$$752$$ − 2.38332i − 0.0869108i
$$753$$ 16.2044i 0.590522i
$$754$$ 15.9974 0.582591
$$755$$ 0 0
$$756$$ 0.501897 0.0182538
$$757$$ − 6.60568i − 0.240088i −0.992769 0.120044i $$-0.961697\pi$$
0.992769 0.120044i $$-0.0383035\pi$$
$$758$$ 0.468553i 0.0170186i
$$759$$ 6.70661 0.243435
$$760$$ 0 0
$$761$$ −22.8638 −0.828814 −0.414407 0.910092i $$-0.636011\pi$$
−0.414407 + 0.910092i $$0.636011\pi$$
$$762$$ − 8.37978i − 0.303568i
$$763$$ 2.31051i 0.0836461i
$$764$$ −56.9656 −2.06094
$$765$$ 0 0
$$766$$ 37.1103 1.34085
$$767$$ − 49.8760i − 1.80092i
$$768$$ 13.1557i 0.474716i
$$769$$ 35.0078 1.26241 0.631206 0.775615i $$-0.282560\pi$$
0.631206 + 0.775615i $$0.282560\pi$$
$$770$$ 0 0
$$771$$ −3.36942 −0.121347
$$772$$ − 54.4016i − 1.95796i
$$773$$ − 17.7019i − 0.636691i −0.947975 0.318346i $$-0.896873\pi$$
0.947975 0.318346i $$-0.103127\pi$$
$$774$$ 8.93586 0.321193
$$775$$ 0 0
$$776$$ −10.0282 −0.359992
$$777$$ − 0.684020i − 0.0245391i
$$778$$ − 26.6655i − 0.956005i
$$779$$ −20.4434 −0.732460
$$780$$ 0 0
$$781$$ −11.1564 −0.399206
$$782$$ 35.7388i 1.27802i
$$783$$ 1.00000i 0.0357371i
$$784$$ −2.95968 −0.105703
$$785$$ 0 0
$$786$$ −22.5066 −0.802782
$$787$$ 5.13176i 0.182928i 0.995808 + 0.0914638i $$0.0291546\pi$$
−0.995808 + 0.0914638i $$0.970845\pi$$
$$788$$ 32.7321i 1.16603i
$$789$$ −1.79671 −0.0639647
$$790$$ 0 0
$$791$$ −0.653321 −0.0232294
$$792$$ − 8.39044i − 0.298141i
$$793$$ − 80.6662i − 2.86454i
$$794$$ 11.2471 0.399145
$$795$$ 0 0
$$796$$ 36.2783 1.28585
$$797$$ 27.5492i 0.975842i 0.872888 + 0.487921i $$0.162245\pi$$
−0.872888 + 0.487921i $$0.837755\pi$$
$$798$$ 1.44630i 0.0511984i
$$799$$ −42.8974 −1.51760
$$800$$ 0 0
$$801$$ 4.33165 0.153051
$$802$$ 82.0562i 2.89751i
$$803$$ − 39.2424i − 1.38484i
$$804$$ 46.9329 1.65520
$$805$$ 0 0
$$806$$ 163.244 5.75002
$$807$$ − 0.117055i − 0.00412054i
$$808$$ − 39.7778i − 1.39938i
$$809$$ 5.51555 0.193917 0.0969583 0.995288i $$-0.469089\pi$$
0.0969583 + 0.995288i $$0.469089\pi$$
$$810$$ 0 0
$$811$$ −17.1711 −0.602959 −0.301479 0.953473i $$-0.597480\pi$$
−0.301479 + 0.953473i $$0.597480\pi$$
$$812$$ − 0.501897i − 0.0176131i
$$813$$ 4.06294i 0.142493i
$$814$$ −31.5129 −1.10453
$$815$$ 0 0
$$816$$ −3.24131 −0.113468
$$817$$ 15.7288i 0.550283i
$$818$$ 36.6355i 1.28093i
$$819$$ −1.12829 −0.0394256
$$820$$ 0 0
$$821$$ 6.53740 0.228157 0.114078 0.993472i $$-0.463609\pi$$
0.114078 + 0.993472i $$0.463609\pi$$
$$822$$ 8.67813i 0.302684i
$$823$$ 35.8588i 1.24996i 0.780641 + 0.624979i $$0.214893\pi$$
−0.780641 + 0.624979i $$0.785107\pi$$
$$824$$ −19.8440 −0.691299
$$825$$ 0 0
$$826$$ −2.56177 −0.0891354
$$827$$ − 2.36177i − 0.0821269i −0.999157 0.0410635i $$-0.986925\pi$$
0.999157 0.0410635i $$-0.0130746\pi$$
$$828$$ − 6.47910i − 0.225164i
$$829$$ −54.0332 −1.87665 −0.938325 0.345754i $$-0.887623\pi$$
−0.938325 + 0.345754i $$0.887623\pi$$
$$830$$ 0 0
$$831$$ −12.5436 −0.435134
$$832$$ − 92.0182i − 3.19016i
$$833$$ 53.2713i 1.84574i
$$834$$ 4.69231 0.162481
$$835$$ 0 0
$$836$$ 40.7000 1.40764
$$837$$ 10.2044i 0.352716i
$$838$$ 76.0663i 2.62767i
$$839$$ −27.6639 −0.955063 −0.477532 0.878615i $$-0.658468\pi$$
−0.477532 + 0.878615i $$0.658468\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 61.1730i 2.10816i
$$843$$ 2.83163i 0.0975266i
$$844$$ 66.4278 2.28654
$$845$$ 0 0
$$846$$ 12.7318 0.437728
$$847$$ − 0.0706761i − 0.00242846i
$$848$$ 5.12721i 0.176069i
$$849$$ −28.2249 −0.968676
$$850$$ 0 0
$$851$$ −8.83017 −0.302694
$$852$$ 10.7779i 0.369245i
$$853$$ − 16.7180i − 0.572413i −0.958168 0.286207i $$-0.907606\pi$$
0.958168 0.286207i $$-0.0923944\pi$$
$$854$$ −4.14324 −0.141779
$$855$$ 0 0
$$856$$ −11.9021 −0.406805
$$857$$ − 50.6304i − 1.72950i −0.502202 0.864750i $$-0.667477\pi$$
0.502202 0.864750i $$-0.332523\pi$$
$$858$$ 51.9804i 1.77458i
$$859$$ 23.0473 0.786363 0.393182 0.919461i $$-0.371374\pi$$
0.393182 + 0.919461i $$0.371374\pi$$
$$860$$ 0 0
$$861$$ −0.819148 −0.0279165
$$862$$ 1.88693i 0.0642692i
$$863$$ − 45.4971i − 1.54874i −0.632733 0.774370i $$-0.718067\pi$$
0.632733 0.774370i $$-0.281933\pi$$
$$864$$ 6.12646 0.208426
$$865$$ 0 0
$$866$$ −51.1531 −1.73825
$$867$$ 41.3403i 1.40399i
$$868$$ − 5.12156i − 0.173837i
$$869$$ −41.4095 −1.40472
$$870$$ 0 0
$$871$$ −105.507 −3.57498
$$872$$ − 37.3155i − 1.26366i
$$873$$ 3.88355i 0.131438i
$$874$$ 18.6706 0.631543
$$875$$ 0 0
$$876$$ −37.9112 −1.28090
$$877$$ 6.44199i 0.217531i 0.994067 + 0.108765i $$0.0346897\pi$$
−0.994067 + 0.108765i $$0.965310\pi$$
$$878$$ − 54.3124i − 1.83295i
$$879$$ −21.9337 −0.739806
$$880$$ 0 0
$$881$$ −43.3346 −1.45998 −0.729990 0.683458i $$-0.760475\pi$$
−0.729990 + 0.683458i $$0.760475\pi$$
$$882$$ − 15.8107i − 0.532375i
$$883$$ 24.6736i 0.830333i 0.909745 + 0.415166i $$0.136277\pi$$
−0.909745 + 0.415166i $$0.863723\pi$$
$$884$$ −169.197 −5.69072
$$885$$ 0 0
$$886$$ 39.6546 1.33222
$$887$$ 37.1527i 1.24747i 0.781637 + 0.623733i $$0.214385\pi$$
−0.781637 + 0.623733i $$0.785615\pi$$
$$888$$ 11.0472i 0.370718i
$$889$$ 0.591021 0.0198222
$$890$$ 0 0
$$891$$ −3.24930 −0.108856
$$892$$ 29.7247i 0.995255i
$$893$$ 22.4104i 0.749936i
$$894$$ 3.32357 0.111157
$$895$$ 0 0
$$896$$ −2.76723 −0.0924466
$$897$$ 14.5653i 0.486322i
$$898$$ − 9.57108i − 0.319391i
$$899$$ 10.2044 0.340336
$$900$$ 0 0
$$901$$ 92.2847 3.07445
$$902$$ 37.7383i 1.25655i
$$903$$ 0.630241i 0.0209731i
$$904$$ 10.5514 0.350933
$$905$$ 0 0
$$906$$ −45.6291 −1.51593
$$907$$ 7.49457i 0.248853i 0.992229 + 0.124427i $$0.0397091\pi$$
−0.992229 + 0.124427i $$0.960291\pi$$
$$908$$ − 70.3248i − 2.33381i
$$909$$ −15.4045 −0.510934
$$910$$ 0 0
$$911$$ 48.6818 1.61290 0.806449 0.591303i $$-0.201386\pi$$
0.806449 + 0.591303i $$0.201386\pi$$
$$912$$ 1.69332i 0.0560714i
$$913$$ − 8.44143i − 0.279371i
$$914$$ 2.91444 0.0964012
$$915$$ 0 0
$$916$$ 6.37598 0.210668
$$917$$ − 1.58737i − 0.0524197i
$$918$$ − 17.3152i − 0.571486i
$$919$$ 26.1563 0.862816 0.431408 0.902157i $$-0.358017\pi$$
0.431408 + 0.902157i $$0.358017\pi$$
$$920$$ 0 0
$$921$$ −0.461574 −0.0152094
$$922$$ − 54.0746i − 1.78085i
$$923$$ − 24.2293i − 0.797515i
$$924$$ 1.63081 0.0536498
$$925$$ 0 0
$$926$$ 32.0552 1.05340
$$927$$ 7.68484i 0.252403i
$$928$$ − 6.12646i − 0.201111i
$$929$$ 8.15366 0.267513 0.133756 0.991014i $$-0.457296\pi$$
0.133756 + 0.991014i $$0.457296\pi$$
$$930$$ 0 0
$$931$$ 27.8299 0.912089
$$932$$ 34.7509i 1.13830i
$$933$$ − 27.5507i − 0.901968i
$$934$$ −16.0157 −0.524051
$$935$$ 0 0
$$936$$ 18.2222 0.595612
$$937$$ 41.9169i 1.36936i 0.728842 + 0.684682i $$0.240059\pi$$
−0.728842 + 0.684682i $$0.759941\pi$$
$$938$$ 5.41915i 0.176941i
$$939$$ −14.9121 −0.486638
$$940$$ 0 0
$$941$$ 14.4789 0.471998 0.235999 0.971753i $$-0.424164\pi$$
0.235999 + 0.971753i $$0.424164\pi$$
$$942$$ 16.0751i 0.523755i
$$943$$ 10.5746i 0.344356i
$$944$$ −2.99931 −0.0976191
$$945$$ 0 0
$$946$$ 29.0353 0.944020
$$947$$ 1.83838i 0.0597392i 0.999554 + 0.0298696i $$0.00950921\pi$$
−0.999554 + 0.0298696i $$0.990491\pi$$
$$948$$ 40.0048i 1.29929i
$$949$$ 85.2262 2.76656
$$950$$ 0 0
$$951$$ −22.9266 −0.743446
$$952$$ 3.15349i 0.102205i
$$953$$ 49.5140i 1.60391i 0.597382 + 0.801957i $$0.296208\pi$$
−0.597382 + 0.801957i $$0.703792\pi$$
$$954$$ −27.3897 −0.886776
$$955$$ 0 0
$$956$$ −7.03461 −0.227516
$$957$$ 3.24930i 0.105035i
$$958$$ − 52.0403i − 1.68135i
$$959$$ −0.612063 −0.0197646
$$960$$ 0 0
$$961$$ 73.1299 2.35903
$$962$$ − 68.4393i − 2.20657i
$$963$$ 4.60924i 0.148531i
$$964$$ −45.4658 −1.46436
$$965$$ 0 0
$$966$$ 0.748115 0.0240702
$$967$$ 5.90299i 0.189827i 0.995486 + 0.0949137i $$0.0302575\pi$$
−0.995486 + 0.0949137i $$0.969742\pi$$
$$968$$ 1.14144i 0.0366874i
$$969$$ 30.4781 0.979096
$$970$$ 0 0
$$971$$ −35.7964 −1.14876 −0.574380 0.818588i $$-0.694757\pi$$
−0.574380 + 0.818588i $$0.694757\pi$$
$$972$$ 3.13907i 0.100686i
$$973$$ 0.330946i 0.0106096i
$$974$$ 61.2801 1.96354
$$975$$ 0 0
$$976$$ −4.85088 −0.155273
$$977$$ − 1.79673i − 0.0574825i −0.999587 0.0287412i $$-0.990850\pi$$
0.999587 0.0287412i $$-0.00914988\pi$$
$$978$$ 14.9328i 0.477497i
$$979$$ 14.0748 0.449834
$$980$$ 0 0
$$981$$ −14.4509 −0.461382
$$982$$ − 26.1828i − 0.835528i
$$983$$ − 6.24151i − 0.199073i −0.995034 0.0995365i $$-0.968264\pi$$
0.995034 0.0995365i $$-0.0317360\pi$$
$$984$$ 13.2295 0.421742
$$985$$ 0 0
$$986$$ −17.3152 −0.551427
$$987$$ 0.897966i 0.0285826i
$$988$$ 88.3917i 2.81211i
$$989$$ 8.13593 0.258708
$$990$$ 0 0
$$991$$ 41.0410 1.30371 0.651855 0.758344i $$-0.273991\pi$$
0.651855 + 0.758344i $$0.273991\pi$$
$$992$$ − 62.5169i − 1.98491i
$$993$$ 3.23147i 0.102548i
$$994$$ −1.24448 −0.0394725
$$995$$ 0 0
$$996$$ −8.15507 −0.258403
$$997$$ 29.8910i 0.946657i 0.880886 + 0.473329i $$0.156948\pi$$
−0.880886 + 0.473329i $$0.843052\pi$$
$$998$$ − 8.56776i − 0.271208i
$$999$$ 4.27815 0.135355
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2175.2.c.o.349.12 14
5.2 odd 4 2175.2.a.ba.1.2 7
5.3 odd 4 2175.2.a.bb.1.6 yes 7
5.4 even 2 inner 2175.2.c.o.349.3 14
15.2 even 4 6525.2.a.bx.1.6 7
15.8 even 4 6525.2.a.bu.1.2 7

By twisted newform
Twist Min Dim Char Parity Ord Type
2175.2.a.ba.1.2 7 5.2 odd 4
2175.2.a.bb.1.6 yes 7 5.3 odd 4
2175.2.c.o.349.3 14 5.4 even 2 inner
2175.2.c.o.349.12 14 1.1 even 1 trivial
6525.2.a.bu.1.2 7 15.8 even 4
6525.2.a.bx.1.6 7 15.2 even 4